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Representing a function of one variable in terms of a sequence of gaussians

  1. Nov 15, 2014 #1
    Is this possible? It seems like it should be but, it's difficult to find an explicit relationship between a general function of one variable x (let's say we are only interested in functions that decay to zero as they go to plus or minus infinity)

    it seems like summing a bunch of gaussians of arbitrary width located at various points along the domain, one should be able to construct an arbitrary function but I don't see much useful material on it.
  2. jcsd
  3. Nov 15, 2014 #2

    Stephen Tashi

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    Science Advisor

    Look at the topic of "radial basis functions" and think of the 1-dimensional case.

    You also might find the topic of "wavelets" interesting. I don't know whether one can make useful "wavelets" with a gaussian shape.
  4. Nov 16, 2014 #3
    Radial basis functions are the answer if by "representing" you mean "interpolating". Let [itex] f [/itex] be a function and consider the interpolant
    [tex]\sigma(x) = \sum_{j=1}^N a_j e^{-\frac{(x-x_j)^2}{c^2} }, [/tex]
    for some [itex] c [/itex] and some set of nodes [itex] \{x_j\} [/itex]. The interpolation conditions [itex] f(x_j) = \sigma(x_j) [/itex] give you a system of equations for the coefficients [itex] a_j [/itex]. This system always has a solution, because the gaussian is positive definite. However, note that this is not an exact representation, you are just interpolating.
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